﻿178 
  Prof. 
  J. 
  J. 
  Thomson 
  on 
  a 
  Theory 
  of 
  the 
  Connexion 
  

  

  be 
  a 
  sphere 
  of 
  radius 
  d, 
  we 
  shall 
  suppose 
  that 
  the 
  electric 
  

   intensity 
  and 
  the 
  magnetic 
  force 
  both 
  vanish. 
  

  

  Thus 
  the 
  original 
  distribution 
  of 
  the 
  field 
  is 
  confined 
  

   between 
  two 
  parallel 
  planes; 
  and 
  from 
  this 
  space 
  we 
  must 
  

   exclude 
  that 
  inside 
  the 
  sphere 
  as 
  this 
  is 
  free 
  from 
  magnetic 
  

   force. 
  

  

  Let 
  us 
  now 
  consider 
  how 
  this 
  distribution 
  will 
  spread 
  through 
  

   space. 
  Consider 
  what 
  will 
  happen 
  at 
  a 
  point 
  P. 
  There 
  will 
  

   be 
  no 
  effect 
  at 
  P 
  until 
  a 
  sphere 
  of 
  radius 
  V* 
  and 
  centre 
  P 
  

   cuts 
  the 
  space 
  between 
  the 
  planes. 
  This 
  will 
  not 
  happen 
  until 
  

   t=(c—d)/Y, 
  where 
  c 
  is 
  the 
  distance 
  of 
  P 
  from 
  the 
  plane 
  

   through 
  the 
  centre 
  of 
  the 
  sphere 
  perpendicular 
  to 
  the 
  direction 
  

   in 
  which 
  the 
  sphere 
  was 
  moving 
  before 
  it 
  was 
  stopped. 
  When 
  

   t 
  is 
  greater 
  than 
  this 
  value, 
  the 
  sphere 
  will 
  cut 
  the 
  space 
  

   between 
  the 
  planes 
  ; 
  and 
  to 
  apply 
  Poisson's 
  solution 
  we 
  have 
  

   to 
  find 
  the 
  mean 
  value 
  of 
  the 
  magnetic 
  force 
  over 
  the 
  surface 
  

   of 
  this 
  sphere. 
  Take 
  the 
  plane 
  of 
  xz 
  to 
  pass 
  through 
  P. 
  Let 
  

   Q 
  be 
  a 
  point 
  on 
  the 
  surface 
  of 
  the 
  sphere, 
  a?S 
  an 
  element 
  of 
  

   the 
  area 
  of 
  this 
  surface, 
  <p 
  the 
  angle 
  the 
  plane 
  through 
  Q 
  and 
  

   the 
  axis 
  of 
  z 
  makes 
  with 
  the 
  plane 
  of 
  xz, 
  p 
  the 
  distance 
  of 
  Q 
  

   from 
  the 
  axis 
  of 
  z, 
  and 
  6 
  the 
  angle 
  between 
  p 
  and 
  the 
  normal 
  

   to 
  tbe 
  sphere 
  at 
  Q 
  ; 
  then 
  the 
  element 
  of 
  the 
  surface 
  included 
  

   between 
  z 
  and 
  z-\ 
  dz, 
  </> 
  and 
  </> 
  + 
  d<f> 
  is 
  given 
  by 
  the 
  equation 
  

  

  JO 
  dd> 
  dz 
  

  

  db 
  = 
  jr 
  

  

  ' 
  cos 
  

  

  Now 
  initially 
  

  

  /8= 
  

  

  fiV 
  COS 
  (/> 
  m 
  

  

  so 
  

  

  that 
  

  

  pd 
  

   fidS 
  = 
  -, 
  J 
  dcj) 
  dz. 
  

  

  d 
  cos 
  6 
  

  

  Now 
  if 
  a 
  is 
  written 
  for 
  V£ 
  the 
  radius 
  of 
  the 
  sphere, 
  and 
  if 
  

   the 
  x 
  coordinate 
  of 
  P 
  is 
  b, 
  then 
  we 
  may 
  easily 
  prove 
  that 
  

  

  cos0= 
  - 
  >/a 
  s 
  -(«— 
  c) 
  8 
  — 
  6 
  a 
  sin 
  a 
  ^; 
  

  

  hence 
  

  

  _ 
  70 
  eoN 
  cos 
  4> 
  deb 
  dz 
  

  

  nab 
  = 
  — 
  j 
  , 
  n 
  

  

  d 
  s/a 
  2 
  -(z-cy-tfsm 
  2 
  <f> 
  

  

  The 
  limits 
  of 
  <jf> 
  are 
  

  

  , 
  \/a 
  2 
  — 
  (z 
  — 
  c) 
  2 
  

   -fsm- 
  1 
  -^ 
  ^ 
  L 
  = 
  isin- 
  1 
  ^ 
  say, 
  

  

  b 
  

  

  